The transfer function of a linear device describes the behavior of the device in the Laplace transform (s-plane) domain in response to an applied stimulus. In the design of electrical, mechanical or electromechanical devices it is often important to know the location and strength of each pole and zero of the device transfer function in the s-plane so that the device may be optimized for a desired application. In particular, where device stability is important, knowledge of pole locations is critical. In order for the pole and zero measurement to be of use to the device designer, it is important that the measurement be performed quickly and accurately and that errors due to noise and distortion be minimized.
Various apparatus have been constructed according to the prior art which measure the transfer function of a linear device. Exemplary of such prior art apparatus is that disclosed by Edwin A. Sloane in his U.S. Pat. No. 3,973,112 which was issued on Aug. 3, 1976, and that disclosed by Sloane et al in U.S. Pat. No. 4,047,002 which was issued on Sept. 6, 1977. Typical prior art apparatus have provided pole and zero measurements which have been inherently erroneous due t nonlinearities in the response of the device being analyzed and due to other noise on the measured data.
In guidance with the illustrated preferred embodiment of the present invention, a pole and zero analyzer is disclosed which performs the pole and zero measurement quickly and accurately and which corrects for noise-induced errors in the measurement. The pole and zero measurement is initiated by the application of a desired stimulus signal, such as random or Gaussian noise, to the device. The stimulus and response signals are sampled in the time domain and are transformed to the frequency domain via a Fast Fourier Transform so that the auto-spectrum of the stimulus signal and the cross-spectrum of the stimulus and response signals may be measured. In order to minimize pole-zero measurement error caused by noise and non-linearities in the device response, an ensemble of stimulus and response signal measurements may be made and the auto- and cross-spectra may be measured as ensemble averages. The measured transfer function of the device is determined from the cross- and auto-spectra and the noise level on the measured data is estimated from the ensemble averages of the stimulus and response signals. Measurement time delays may be removed as required.
The measured transfer function is fitted to an estimated transfer function comprising a rational fraction of numerator and denominator Chebyshev polynomials in the variable, s. A weighting function is determined which may be used to emphasize certain portions of the transfer function for the purpose of increasing the accuracy of the pole and zero measurements. The coefficients of the numerator and denominator polynomials are then found as the weighted least squares fit of the measured transfer function data to the estimated transfer function. The quality of the fit of the estimated transfer function to the measured transfer function is determined. If the fit is insufficient, the orders of the numerator and denominator polynomials are varied, new coefficients are determined and the fit is again tested. When a sufficient fit is achieved, the Chebyshev numerator and denominator polynomials of the estimated transfer function are converted to ordinary polynomials and a root solver is used to find the roots of the two polynomials which yield the poles and zeroes of the estimated transfer function. The poles and zeroes may be displayed as desired by the device designer.